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computer aided medical procedures & augmented reality | campar.cs.tum.edu Inter and Intra-Modal Deformable Registration: Continuous Deformations Meet Efficient Optimal Linear Programming Ben Glocker 1,2 , Nikos Komodakis 1,3 , Nikos Paragios 1 , Georgios Tziritas 3 , Nassir Navab 2 4 July 2007 1 GALEN group | laboratoire de mathématiques appliquées aux systèmes | ecole centrale paris 2 computer aided medical procedures & augmented reality | technische universität münchen 3 computer science department | university of crete

computer aided medical procedures & augmented reality | campar.cs.tum.edu Outline � Introduction & Motivation � Image Registration based on Discrete Labeling � Optimization using Linear Programming � Results & Conclusions In this presentation everything is intensity-based camp+ar | department of computer science | technische universität münchen | 11 July 2007 2

computer aided medical procedures & augmented reality | campar.cs.tum.edu Introduction � Source and target image f : Ω → R Ω ⊂ R d with d ∈ { 2 , 3 } g : Ω → R � Image relation g ( x ) = h ◦ f ( T ( x )) T : Ω → Ω non-linear transformation h : R → R non-linear relation on intensities camp+ar | department of computer science | technische universität münchen | 11 July 2007 3

computer aided medical procedures & augmented reality | campar.cs.tum.edu Registration as an Optimization Problem � Energy formulation Z E ( T ) = ρ h ( g ( x ) , f ( T ( x )) d x → min! Ω ρ h : R × R → R distance measure � The aim of registration is to recover the transformation which involves � the definition of a transformation type � the definition of a distance/similarity measure � the definition of an optimization procedure camp+ar | department of computer science | technische universität münchen | 11 July 2007 4

computer aided medical procedures & augmented reality | campar.cs.tum.edu Review of Registration Methods � Types of transformations � Rigid, affine, projective � Basis functions, Spline-based � Finite Element Models, … � Distance/Similarity measures � SAD, SSD, NCC, NMI, CR, … � Optimization methods � Variational � Gradient-based � Direct search (Simplex, Powell-Brent, Best Neighbor) camp+ar | department of computer science | technische universität münchen | 11 July 2007 5

computer aided medical procedures & augmented reality | campar.cs.tum.edu Motivation � What is expected from an optimal registration method? � Independent from the choice of the transformation type � Independent from the choice of the distance/similarity measure � Guarantee of a globally optimal solution � Reasonable computational complexity camp+ar | department of computer science | technische universität münchen | 11 July 2007 6

computer aided medical procedures & augmented reality | campar.cs.tum.edu Our Contributions � Novel deformable registration framework based on discrete labeling and linear programming � Our framework bridges the gap between continuous deformations and discrete optimization � Gradient-free and flexible in the choice of the distance measure � Guaranteed optimality properties on the solution � Computational efficient and tractable camp+ar | department of computer science | technische universität münchen | 11 July 2007 7

computer aided medical procedures & augmented reality | campar.cs.tum.edu Image Registration based on Discrete Labeling camp+ar | department of computer science | technische universität münchen | 11 July 2007 8

computer aided medical procedures & augmented reality | campar.cs.tum.edu Local Registration � Deformation grid providing a continuous and dense deformation field T ( x ) = x + D ( x ) X with D ( x ) = η ( | x − p | ) d p p ∈ G � In our implementation we use Free Form Deformation X 3 X 3 D ( x ) = B l ( u ) B m ( v ) d i + l,j + m m =0 l =0 [Rueckert99, Schnabel01, Rohlfing03, …] camp+ar | department of computer science | technische universität münchen | 11 July 2007 9

computer aided medical procedures & augmented reality | campar.cs.tum.edu Energy Formulation � Reformulation of the optimization problem Z X η − 1 ( | x − p | ) · ρ h ( g ( x ) , f ( T ( x ))) d x E data ( T ) = Ω p ∈ G � Smoothness term X E smooth ( T ) = φ ( | ∇ G d p | ) p ∈ G � Registration task E total = E data + E smooth → min! camp+ar | department of computer science | technische universität münchen | 11 July 2007 10

computer aided medical procedures & augmented reality | campar.cs.tum.edu V pq ( u p , u q ) Discrete Optimization Problem p p q q r r V p ( u p ) � General energy formulation s s t t u u E total = E data + E smooth → min! v v w w x x � Markov Random Field (MRF) formulation for discrete labelings X X E total ( u ) = V p ( u p ) + V pq ( u p , u q ) p ∈ G p,q ∈ E Data term = singleton potentials Smoothness term = pairwise potentials camp+ar | department of computer science | technische universität münchen | 11 July 2007 11

computer aided medical procedures & augmented reality | campar.cs.tum.edu Discretization of Parameter Space � Set of labels and a discretized deformation space L = { u 1 , ..., u i } Θ = { d 1 , ..., d i } +Y +Y Max Steps -X +X -X +X -Y -Y Sparse sampling Dense sampling camp+ar | department of computer science | technische universität münchen | 11 July 2007 12

computer aided medical procedures & augmented reality | campar.cs.tum.edu Data Term � |L| × |G| cost matrix MRF singleton potentials: Z X η − 1 ( | x − p | ) ρ h ( g ( x ) , f ( T ( x ))) d x E data ( u ) = Ω p ∈ G X ≈ V p ( u p ) p ∈ G � Problem: singleton potentials are not independent! camp+ar | department of computer science | technische universität münchen | 11 July 2007 13

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=0 Current label y=0 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 14

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=10 Current label y=0 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 15

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=10 Current label y=-10 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 16

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=0 Current label y=-10 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 17

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=-10 Current label y=-10 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 18

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=-10 Current label y=0 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 19

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=-10 Current label y=10 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 20

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=0 Current label y=10 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 21

computer aided medical procedures & augmented reality | campar.cs.tum.edu Fast Approximation of Singleton Potentials � Approximation of label costs simultaneously for all nodes Nodes Labels x=10 Current label y=10 Single potential look-up table camp+ar | department of computer science | technische universität münchen | 11 July 2007 22